Any triangle without a right angle is called an oblique ...



Any triangle without a right angle is called an oblique triangle. As usual, we use (, (, ( for the angles of the triangle (or the vertices A, B, and C) and a, b, and c, respectively, for the lengths of the sides opposite those angles.

The Law of Sines: In any triangle, the ratio of the sine of an angle to the side opposite that angle is equal to the ratio of the sine of another angle to the side opposite that angle.

The Law of Sines:

If ABC is an oblique triangle labeled in the usual manner, then:

[pic]

Solve (ABC : (This means given 3 pieces of data about a triangle, find the remaining 3 pieces of data.)

( = 41(, ( = 77(, a = 10.5

|a |10.5 |

|b | |

|c | |

|[pic] |41.0° |

|[pic] | |

|[pic] |77.0° |

[pic]

( = 81(, c = 11, b = 12

Draw Triangle:

Review:

Calculate

sin(10() and sin(170() sin(80() and sin(100()

sin(30() and sin(150() sin(1() and sin(179()

Notice that the sines of supplementary angles are equivalent.

Find the angle:

sin(x) = 0.3456 sin(x) = 0.9876

There are two angles that make each equation a true statement, one acute and one obtuse. This may or may not lead to two different triangles.

When you are given one angle and two sides, and inverse sine is used to calculate an angle, there is a possibility of two triangles existing for the given information. This happens when there is SSA. Remember that the sum of the angles in a triangle is 180(. [pic]

( = 53(, a = 140, c = 115 Draw Δ1: Draw Δ 2:

|a |140.0 |140.0 |

|b | | |

|c |115.0 |115.0 |

|A | | |

|B | | |

|C |53.0° |53.0° |

[pic]

( = 75(, b = 248, c =195 Draw Δ:

|a | |

|b |248.0 |

|c |195.0 |

|A | |

|B |75.0° |

|C | |

[pic]

Fire lookout: A fire is spotted from two fire lookouts stations. If station B reports the fire at S37(E, and station A, 10 miles due East of station B, reports the fire at S61(W, how far is the fire from station A? How far is the fire from station B?

|From Point B |4.8 |Miles |

|From Point A |8.1 |Miles |

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Finding the Height of an Airplane. An aircraft is spotted by two observers who are 1000 feet apart. As the airplane passes between them and over the line joining them, observer A sights a 40( angle of elevation to the plane, and observer B sights a 35( angle. How high is the airplane?

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Finding the Length of a Ski Lift.

a) What is the length of the ski lift? b) What is the height of the mountain?

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Surveying A surveyor notes that the direction from point A to point B is N78(E and the direction from A to C is N21(E. The distance from A to B is 345 meters and the distance from B to C is 543 meters. Approximate the distance from A to C.

[pic]

-----------------------

B

a

C

[pic]

c

[pic]

b

A

[pic]

Always draw a triangle.

This triangle does not exist. The side opposite the 81° angle must be longer than 11.

[pic]

Final Answers: The largest side should always be opposite the largest angle. The smallest side should always be opposite the smallest angle.

Notice: The angles given for the measures are outside of the triangle formed from the fire to A to B.

a

b

There are 3 triangles in this picture, the big one and two right triangles.

Again, there are 3 triangles, the big one and two smaller ones, a left one and a right. In the left triangle, angle A would be 155° and the very small angle up by B is 10°.

25(

15(

1000 ft

A

D

B

C

Mountain

Ski Lift

57°

Note: This picture may not be to scale.

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